# L9n20

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n20 at Knotilus! L9n20 is $9^3_{16}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 u v w^2+2 u v w-u v+u w^2-u w+v w^2-v w+w^3-2 w^2+2 w}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $-q^8+2 q^7-4 q^6+5 q^5-4 q^4+6 q^3-3 q^2+3 q$ (db) Signature 2 (db) HOMFLY-PT polynomial $-2 z^4 a^{-4} +3 z^2 a^{-2} -7 z^2 a^{-4} +3 z^2 a^{-6} +5 a^{-2} -10 a^{-4} +6 a^{-6} - a^{-8} +2 a^{-2} z^{-2} -5 a^{-4} z^{-2} +4 a^{-6} z^{-2} - a^{-8} z^{-2}$ (db) Kauffman polynomial $z^7 a^{-5} +z^7 a^{-7} +4 z^6 a^{-4} +6 z^6 a^{-6} +2 z^6 a^{-8} +3 z^5 a^{-3} +5 z^5 a^{-5} +3 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-4} -16 z^4 a^{-6} -5 z^4 a^{-8} -6 z^3 a^{-3} -18 z^3 a^{-5} -15 z^3 a^{-7} -3 z^3 a^{-9} +6 z^2 a^{-2} +18 z^2 a^{-4} +15 z^2 a^{-6} +3 z^2 a^{-8} +11 z a^{-3} +21 z a^{-5} +13 z a^{-7} +3 z a^{-9} -7 a^{-2} -14 a^{-4} -10 a^{-6} -2 a^{-8} -5 a^{-3} z^{-1} -9 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1} +2 a^{-2} z^{-2} +5 a^{-4} z^{-2} +4 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
01234567χ
17       1-1
15      1 1
13     31 -2
11    21  1
9   34   1
7  31    2
5  3     3
333      0
13       3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.